Peridynamics Research Articles

Peridynamic Framework to Model Additive Manufacturing Processes

AbstractThe study presents a framework for analyzing Additive Manufacturing processes within the Peridynamics (PD) software PeriLab. This framewor k employs a mesh‐free, point‐based numerical approach to approximate the continuum PD equations. Implemented within this framework are thermal, thermo‐mechanical, and simple additive models. These models have been validated against analytical solutions, Finite Element (FE) models, and Peridigm simulations. To leverage the PD mesh‐free implementation, the study introduces a novel boundary detection algorithm. This algorithm is essential because the outer surface area may change during the manufacturing process. It operates without requiring surface or topology information, relying instead on the comparison of neighborhood volume to sphere volume. Additionally, the study introduces a wrapper that generates the mesh necessary for simulating the printing process, based on the G‐code machine input path. Finally, the study presents a comprehensive analysis of an L‐shaped profile utilizing the developed features, comparing the results with those obtained from an Abaqus solution.

Peridynamics simulations of the damage of reinforced concrete structures under radial blasting

Concrete is prone to damage under explosive loads, which can cause a large number of casualties and property losses. The concrete fragmentation process during explosion is transient and dynamic, and the experimental measurement of such events is difficult and risky to conduct, and the intermediate explosion process is difficult to observe in the experimental tests. Therefore, the numerical simulation is an ideal method to model and simulate the explosion process of concrete. Different from the traditional finite element method, Peridynamics (PD) method uses the spatial integral equation to replace the traditional local differential equation to solve the fragmentation problem with massive and complex discontinuous patterns. In this study, a peridynamics (PD) model is developed to simulate the failure process of reinforced concrete (RC) structures under radial blasting. Concrete PD models with different cavity sizes and reinforcement conditions were established and calibrated with the experimental data. We find that the crack growth and damage pattern obtained in the peridynamics simulation is consistent with the experiment test results, which verifies the feasibility of peridynamics method as a modeling tool for modeling concrete damage under explosive load and for evaluating anti-explosion performance of RC concrete structures.

A coupled peridynamics–smoothed particle hydrodynamics model for fluid–structure interaction with large deformation

Fluid–structure interaction (FSI) is ubiquitous in various engineering disciplines, and effectively managing FSI often appears to be the key for successful failure analysis and safety-oriented design. Smoothed particle hydrodynamics (SPH) serves as a potent nonlocal meshfree method for fluid dynamics modeling, while peridynamics (PD) demonstrates exceptional capability in addressing structural dynamics involving large deformations and discontinuities. Thus, leveraging their respective strengths in a combined approach holds significant promise for tackling FSI challenges. In this work, we propose a new peridynamics–smoothed particle hydrodynamics (PD-SPH) coupling model for addressing FSI. A stable and efficient coupling algorithm for data transfer between PD and SPH is put forward. In this coupling strategy, a PD particle directly participates in solving the SPH governing equations when it is identified to be within the support domain of an SPH particle. This can be done since the SPH quantities including the density, velocity, and pressure of a PD particle are naturally attainable within the framework of non-ordinary state-based peridynamics theory. Concurrently, in solving PD governing equations, reaction forces from SPH particles act as external forces for PD particles, determined straightforwardly through Newton's third law. As such, the proposed PD-SPH coupling strategy is straightforward to implement and offers high computational efficiency. Validation examples demonstrate that the proposed PD-SPH coupling model is computationally robust and adept at capturing physical phenomena in diverse FSI scenarios involving breaking free surfaces of fluid and large structural deformations of solid. Moreover, the proposed PD-SPH coupling model is flexible introducing no constraint conditions for applications and can accommodate different particle resolutions for PD and SPH domains. These features enable a broad application range of the proposed PD-SPH coupling model including simulations of explosion-induced soil fragmentation, rock fracture, and concrete dam failure, which will be conducted by authors in the near future.

Dynamic tensile behaviors and crack propagation in fiber-reinforced cementitious composites: Experimental investigation and Peridynamic simulation

Fiber-Reinforced Cementitious Composites (FRCC) have gained significant attention in engineering applications due to their superior mechanical properties and toughness, particularly under high strain rate conditions. This study performed dynamic tensile tests on FRCC at high strain rates (35-110 s⁻¹) using a Split Hopkinson Tensile Bar (SHTB) apparatus. Additionally, a novel Peridynamic (PD) model was developed for the SHTB and FRCC system, leveraging the advanced capabilities of the emerging PD theory. The study compared and analyzed dynamic tensile strength, ultimate tensile strain, strain rate effects, failure modes, and crack development in FRCC with different fiber ratios at various high strain rates, using both experimental data and PD simulations. The results show that the PD-SHTB-FRCC dynamic model developed in this study exhibits high consistency between numerical simulations and experimental findings, effectively capturing the processes of crack initiation, propagation, and complete failure in FRCC specimens. The dynamic tensile properties of FRCC improve significantly with increased strain rates, with polyethylene (PE) fibers providing superior reinforcement compared to steel fibers. Notably, the dynamic tensile strength, peak tensile stress, and ultimate tensile strain of FRCC increase significantly with rising strain rates, with specimens containing higher PE fiber content showing a more pronounced enhancement effect. For strain rates between 42.6 s⁻¹ and 76.1 s⁻¹, considering dynamic tensile strength, ultimate tensile strain, and peak tensile stress, the optimal combination for resisting dynamic tensile loads was 1.5% PE fibers and 0.5% steel fibers. At a strain rate of 99.8 s⁻¹, a 2% PE fiber ratio alone provided the best performance.

A novel hybrid PD-FEM-FVM approach for simulating hydraulic fracture propagation in saturated porous media

To enhance the computational efficiency of fluid–solid coupling in peridynamics (PD), a hybrid modeling approach based on the classical Biot theory is proposed for simulating hydraulic crack propagation in saturated porous media. The deformation and damage of solids are described by the coupling of the finite element method (FEM) and PD. Based on Darcy’s law, the finite volume method (FVM) is used to describe fluid seepage and calculate pore water pressure. The mutual transfer of fluid pressure and solid deformation is realized through the transition layer between the solid layer and the fluid layer. Firstly, the effectiveness of the proposed method is verified by a porous media seepage simulation example. Secondly, the ability and efficiency of this method to simulate crack propagation in saturated porous media are verified by several examples of hydraulic fracturing of rock with a single pre-existing crack. Finally, the synchronous hydraulic fracturing process of rock with double cracks is simulated. The ability of this method to simulate the simultaneous propagation of multiple fractures in the rock under fluid–solid coupling is further illustrated. The aforementioned studies demonstrate that the novel hybrid PD-FEM-FVM approach not only ensures computational accuracy and effectiveness but also significantly enhances computational efficiency.

Damage and Crack Propagation Mechanism of Q345 Specimen Based on Peridynamics with Temperature and Bolt Holes

With the increasing demand for the performance and design refinement of steel structures (including houses, bridges, and infrastructure), many structures have adopted ultimate bearing capacity in service. The design service lives of steel building structures are generally more than 50 years, and most of them contain bolted connections, which suffer from extreme conditions such as fire (high temperature) during service. When the structure contains defects or cracks and bolt holes, it is easy to produce stress concentration at the defect location, which leads to crack nucleation and crack propagation, reduces the bearing capacity of the structure, and causes the collapse of the structure and causes disasters. In the process of structural damage and crack propagation, the traditional method has some disadvantages, such as stress singularity, the mesh needing to be redivided, and the crack being restricted to mesh; however, the integral method of peridynamics (PD) can completely avoid these problems. Therefore, in this paper, the constitutive equation of PD in high temperature is derived according to the variation law of steel material properties when changed by temperature increase and peridynamics parameters; the damage and crack expansion characteristics of Q345 steel specimens with bolt holes and a central double-crack at 20 °C, 200 °C, 400 °C, and 600 °C were analyzed to clarify the structural damage and failure mechanism. This study is helpful for providing theoretical support for the design of high-temperature steel structures, improving the stability of the structure, and ensuring the bearing capacity of the structure and the safety of people’s lives and property.

Bond-associated non-ordinary state-based peridynamics for simulating damage evolution of thermal barrier coatings in aero-engine turbine blades

The failure mode of thermal barrier coatings (TBC) systems in aeroengine turbine blades is very complex because of the harsh service conditions. A peridynamic (PD) model is established to simulate the damage evolution of TBC with uniform thermally grown oxide (TGO) growth under cycle load. The peridynamic differential operator is introduced to solve the zero-energy mode, and thermo-elastic deformation is considered. Moreover, the influence of high-temperature holding time, initial oxide layer thickness, and interface morphology on the evolution of the stress distribution and interface damage is discussed. The newly proposed PD model can effectively capture the interface cracking of TBC systems and it is conducive to the study of the failure of TBC systems.

Peridynamics model of torsion-warping: Application to lattice beam structures

This paper presents a finite deformation beam model based on Simo-Reissner theory in peridynamics (PD) framework to deal with torsion induced warping deformation. Seven degrees of freedom, viz. three translational, three rotational, and one warping amplitude are considered at each material point. The governing equations of the beam are obtained by employing global balance of linear and angular momenta in conjunction with Simo's assumption on the deformation field. The relation between PD resultant force, moment, bi-moment, and bi-shear states with their classical counterparts is established using the constitutive correspondence method. Numerical implementation strategy is furnished for both quasi-static and dynamic cases. The solution for quasi-static load is obtained through the Newton-Raphson method. The proposed model is validated against finite element solutions considering cantilever beam and lattice structures. Quasi-static deformation responses of 3×3×3 octet and single unit compression-torsion lattice structures are presented further to demonstrate the effectiveness of proposed beam model. A new bond breaking criterion is proposed based on critical stretch, critical relative rotation, and critical relative warping amplitude and failure of the compression-torsion lattice structures under compressive load is simulated. The Newmark-beta method is utilized to solve the governing equations for dynamic loading. Numerical simulations include dynamic analysis of octet and compression-torsion lattice structures.

A peridynamic-based reconstruction and analysis method for short-fiber reinforced composites

Abstract As the complex microstructure, predicting the effective elastic properties and damage mechanisms of Short-Fiber Reinforced Composites (SFRC) continues to pose a challenge. This study presents a new reconstruction and analysis method based on PeriDynamics (PD) for SFRC. The reconstruction of the surface profiles of short-fibers obtained from Micro-CT (Micro-Computed Tomography) scanning resulted in a PD computational model that accurately reflects the actual microstructure of SFRC. The results predicted the effective elastic modulus of SFRC while fiber volume fraction and distribution have impact. Micro-cracks emerge at stress concentration regions near the fiber-matrix-fiber interfaces between two adjacent fibers, and with increasing load, these cracks gradually propagate, interact, and merge with each other. Eventually, major cracks form, leading to material failure.

Coupled field modelling of thermomechanical response in materials using peridynamic heat transport model and non-ordinary state-based peridynamics

Predicting the mechanical response of materials subjected to transient heat processes is crucial in various engineering applications. This paper introduces a novel peridynamic (PD) coupled field model formulated within the framework of Non-Ordinary State-Based Peridynamics (NOSBPD). This model offers the capability to simulate the coupled thermo-mechanical behavior of materials by incorporating both thermal transport and mechanical deformation analyses. To verify the model’s accuracy, a benchmark problem involving a steel plate undergoing transient thermal expansion is investigated. This model presents a valuable tool for various engineering applications involving coupled thermo-mechanical processes.

Peridynamic analysis of rolling contact fatigue crack propagation in rail welding joints with pore defects

Pore defects are prevalent in rail welding joints and significantly contribute to the propagation of fatigue cracks. This study develops a peridynamic (PD) model that incorporates the characteristics of pore defects to analyze their impact on rolling contact fatigue behavior. Initially, compact tension (CT) fatigue tests were performed to derive and validate the bond fatigue failure model specific to rail weld materials. Subsequently, pore defects were modeled as holes in the CT specimens, with experimental results being compared to PD simulation outcomes for validation. Finally, a wheel-rail contact PD model was constructed to investigate the mechanisms of fatigue crack propagation in rail welding joints affected by pore defects.

Peridynamic modelling of elastic and viscoelastic behaviour in polycrystalline ice: A study using NOSBPD and PDCHT

This study explores the elastic and viscoelastic behaviour of ice using the Non-Ordinary State-Based Peridynamic (NOSBPD) framework. A primary objective is to extend and validate the applicability of NOSBPD in modelling the complex responses of ice under various conditions. The study employs the Peridynamic Computational Homogenization Theory (PDCHT) to determine the critical threshold of grain count necessary to induce an effectively isotropic response in polycrystalline S2 ice. Results are consistent with previous findings from Finite Element Method (FEM) and Bond-Based Peridynamics (BBPD) studies.Furthermore, the viscoelastic response of ice is investigated by integrating a viscoelastic constitutive model into the NOSBPD framework. A benchmark problem of a viscoelastic ice sample subjected to tensile stress is simulated, with results compared against FEM simulations conducted in ANSYS Mechanical. The findings show good agreement, validating the NOSBPD framework's capability to capture time-dependent viscoelastic behaviour of ice accurately.The study contributes to the field of ice mechanics by demonstrating the robustness and versatility of NOSBPD in modelling both elastic and viscoelastic responses of ice. These advancements enhance the credibility and applicability of peridynamics (PD) as a powerful tool for simulating complex material behaviours, paving the way for further research and practical applications in ice engineering.

Damage analysis of OC4 jacket subjected to cyclic loading by peridynamic approach

Catastrophic structural failure caused by fatigue damage under cyclic loads can be avoided by identifying critical locations of the damage initiation and the fatigue life during the design stage. Peridynamic (PD) theory defines structure as a collection of material points with non-local bond interactions where the structural discontinuity due to fatigue represented by instantaneous bond breakage is estimated through cumulative decrement of the bond’s life at each load cycle. In this work, we model a reference jacket presented under the Offshore Code Comparison Collaboration Continuation project (OC4) through peridynamic beam formulation. Initially, the static deformations of the beam and deformations of the OC4 jacket under static, harmonic, and irregular point loads are validated with ABAQUS results in all six degrees of freedom. Thereby, the PD fatigue parameters of the jacket’s steel material are calibrated from the experimental data of the corresponding material with a trial simulation for fatigue damage analysis. Later, different load cases from regular waves interacting with the jacket are generated in the PD framework by adopting linear wave theory. Based on the PD cyclic energy release rate model, a comparative study of all load cases to identify critical damage locations with the failure load cycles for damage initiation and fracture is performed for the considered OC4 jacket.

A modified bond-based peridynamic approach for rigid projectile perforation on concrete slabs

Peridynamic (PD) has a unique advantage in describing the crack growth and fragmentation of brittle materials. Concerning the dynamic behaviors and failure patterns of concrete slabs under projectile perforations, a modified bond-based PD approach maintaining both the easy implementation and computational stability characteristics was firstly developed from the following three aspects, (i) a rate-dependent PD constitutive model was proposed for describing the dynamic behaviors of concrete; (ii) a progressive damage criterion considering the tension-compression anisotropy, softening behavior, and strain rate effect of concrete was incorporated to more accurately reproduce the damage and failure of concrete; (iii) an improved micro-modulus function related to bond length was introduced to reveal the internal length effect of bond force. Then, numerical simulations of projectile perforation on concrete slabs by utilizing the developed modified bond-based PD approach, as well as the corresponding sensitivity analyses of discretization parameters including horizon size and particle spacing were performed. Based on the recommended horizon size and particle spacing, the predicted residual velocity of projectile and failure patterns of concrete slabs exhibited an excellent agreement with the test data. Furthermore, by comparisons of the traditional bond-based PD and classical finite element methods, the superiority of developed approach in describing the perforation damage of concrete targets against projectile impact was demonstrated. Finally, the modified bond-based PD approach was employed to blind simulate the projectile normal and oblique perforating multi-layered spaced concrete target plates. It was found that the modified PD model reasonably predicted the terminal ballistic trajectory, deflection angle, and residual velocity of projectile, as well as the failure patterns of target plates. The present work provides a new way to predict the terminal ballistic effect of projectile and dynamic behaviors of concrete slabs.

Finite deformation peridynamics shell theory: Application to mechanical metasurfaces

A finite deformation Simo-Reissner peridynamics (PD) shell theory is developed to investigate wave propagation and crack growth in shell structures. Simo-Reissner deformation approximation is utilized to derive the governing equations of the proposed theory from the three-dimensional PD equation. Deformation states for the PD shell theory are identified from the reduced stress power equation. Classical shell constitutive equations are seamlessly incorporated in PD framework by constitutive correspondence approach where a particular form of the classical reduced stress power expression suitable for constitutive correspondence is used. Rotation tensor is updated using an exponential map. A new bond breaking criterion is proposed for PD shell based on critical stretch and critical relative rotation. The solutions of the PD shell governing equations for quasi-static and dynamic cases are obtained through the Newton-Raphson method and the Newmark-beta method, respectively. Numerical simulations include finite deformation of cylindrical shell under both quasi-static and dynamic loading conditions. The PD shell model is validated against finite element solutions obtained using ANSYS®. Crack propagation through the cylindrical shell is simulated under quasi-static and dynamic loading. Wave propagation through cylindrical shell embedded with periodically arranged holes and cracks is investigated. Significant wave attenuation and large bandgap demonstrates the efficacy of the present proposal.

Peridynamics-fueled convolutional neural network for predicting mechanical constitutive behaviors of fiber reinforced composites

Despite advancements in predicting the constitutive relationships of composite materials, characterizing the effects of microstructural randomness on their mechanical behaviors remains challenging. In this study, we propose a data-driven convolutional neural network (CNN) to efficiently predict the stress-strain curves containing three key material features (Tensile strength, modulus, and toughness) of fiber reinforced composites. Firstly, stress-strain curves for composites with arbitrary fiber distributions were generated using experimentally validated peridynamics (PD) model. Principal component analysis (PCA) was then employed to learn these curves in a lower-dimensional space, reducing computational costs. Subsequently, these reduced data, along with randomly distributed microstructural features, were used to train, validate, and evaluate the CNN models. The combined CNN and PCA model accurately predicted stress-strain curves with maximum errors of 2.5 % for tensile strength, 10% for modulus, and 20 % for toughness. Furthermore, data augmentation and Mean Squared Error (MSE) as a loss function significantly enhanced the model's prediction accuracy. Our findings indicated that DenseNet121 outperformed other CNN models in predicting the properties of fiber-reinforced materials, further demonstrating the effectiveness of the proposed model. This work successfully demonstrates the applicability of a data-driven CNN approach to predict stress-strain relations for engineering materials with intricate heterogeneous microstructures, paving the way for data-driven computational mechanics applied in composites.

Peridynamic modelling of cryogenic deuterium pellet fragmentation for shattered pellet injection in tokamaks

Abstract Shattered pellet injection (SPI) is a promising method for controlling plasma disruptions in tokamaks. In this study, we present numerical modelling of the fragmentation of cryogenic deuterium pellets within the context of SPI, using the peridynamic (PD) theory. A dedicated in-house code has been developed, leveraging the meshfree method and GPU parallelization. The mechanical properties of cryogenic solid deuterium are obtained from available literature, and calibrated based on the shatter threshold along with the remaining solid mass fraction after shatter. The results from the bond-based peridynamics (BB-PD) successfully reproduce the main experimental results reported in the literature, both qualitatively and quantitatively.

Peridynamic study on thermomechanical damage of the rail during wheel idling

Wheel idling is an extreme sliding contact scenario that accelerates rail failure. Using temperature-dependent elastoplastic materials, a peridynamic (PD) model for wheel-rail contact was developed to investigate the thermomechanical damage behavior of rails during wheel idling. Numerical simulations of loading and unloading during wheel idling under different loads were performed, and the results of contact stress, temperature, plastic deformation, and material damage were obtained and analyzed. During loading, the temperature of the rail surface increases rapidly to over 1200 °C, which is well above the austenitization temperature. The intense thermal softening caused by high temperatures deteriorates the wheel-rail contact relationship and leads to a surge in plastic deformation. The softened surface material was continuously removed by the contact load, leading to severe wear damage, and the wear depth increased significantly with the load and idling time. The wear coefficients for the Archard wear model were derived using the wear results from the PD model. During unloading, the surface material cools rapidly at a rate of 1400°C/s through heat transfer into the rail, undergoing martensitic transformation and forming a brittle and hard white etching layer. The thickness of the WEL decreases as the wear depth increases.

Superposition‐based concurrent multiscale approaches for porodynamics

AbstractThe current study presents superposition‐based concurrent multiscale approaches for porodynamics, capable of capturing related physical phenomena, such as soil liquefaction and dynamic hydraulic fracture branching, across different spatial length scales. Two scenarios are considered: superposition of finite element discretizations with varying mesh densities, and superposition of peridynamics (PD) and finite element method (FEM) to handle discontinuities like strain localization and cracks. The approach decomposes the acceleration and the rate of change in pore water pressure into subdomain solutions approximated by different models, allowing high‐fidelity models to be used locally in regions of interest, such as crack tips or shear bands, without neglecting the far‐field influence represented by low‐fidelity models. The coupled stiffness, mass, compressibility, permeability, and damping matrices were derived based on the superposition‐based current multiscale framework. The proposed FEM‐FEM porodynamic coupling approach was validated against analytical or numerical solutions for one‐ and two‐dimensional dynamic consolidation problems. The PD‐FEM porodynamic coupling model was applied to scenarios like soil liquefaction‐induced shear strain accumulation near a low‐permeability interlayer in a layered deposit and dynamic hydraulic fracturing branching. It has been shown that the coupled porodynamic model offers modeling flexibility and efficiency by taking advantage of FEM in modeling complex domains and the PD ability to resolve discontinuities.

Finite Element Method-Peridynamics Coupled Analysis of Slope Stability Affected by Rainfall Erosion

Rainfall is a pivotal factor resulting in the cause of slope instability. The traditional finite element method often fails to converge when dealing with the strongly nonlinear fluid–solid coupling problems, making it impossible to fully analyze the sliding process under the state of slope instability. Therefore, this paper uses the coupling of peridynamics (PD) and the finite element method (FEM) to propose a data exchange mode between the seepage field and the deformation field. The influencing factors of fine particle erosion during rainfall are further considered. According to the damage mechanism of the slope sliding process to the original structure of the soil, a modified erosion constitutive relationship is proposed, which takes into account the destructive effect of plastic deformation on coarse particles. Then, the influence of rainfall duration, rainfall intensity, erosion, and initial saturated permeability coefficient on slope stability was simulated and analyzed. This paper provides a novel concept for slope stability analysis and safety evaluation under rainfall conditions.